International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by Th. Hahn

International Tables for Crystallography (2006). Vol. A. ch. 9.1, p. 749

Section 9.1.9. Example

H. Burzlaffa and H. Zimmermannb*

a Universität Erlangen–Nürnberg, Robert-Koch-Strasse 4a, D-91080 Uttenreuth, Germany, and bInstitut für Angewandte Physik, Lehrstuhl für Kristallographie und Strukturphysik, Universität Erlangen–Nürnberg, Bismarckstrasse 10, D-91054 Erlangen, Germany
Correspondence e-mail:  helmuth.zimmermann@knot.uni-erlangen.de

9.1.9. Example

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This example is discussed in Azároff & Buerger (1958[link], pp. 176–180).

The lattice parameters are given as [b_{1} = 4.693], [b_{2} = 4.936], [b_{3} = 7.524\ \hbox{\AA}], [\beta _{23} = 131.00], [\beta _{31} = 89.57], [\beta _{12} = 90.67^\circ]. The scalar products resulting from these data are given in Table 9.1.9.1[link]. The scalar product [{\bf b}_{1}\cdot{\bf b}_{3}] is positive. Thus the transformation [{\bf b}_{1}' = - {\bf b}_{1}, \ {\bf b}_{3}' = {\bf b}_{3}, \ {\bf b}_{2}' = {\bf b}_{1} + {\bf b}_{2}, \ {\bf b}_{4}' = {\bf b}_{1} + {\bf b}_{4}] is applied. The new scalar products are all non-positive as given in the second row of Table 9.1.9.1[link] (within the accuracy of the experimental data). Comparison with Table 9.1.8.1[link] leads to M6, Voronoi type IV and the monoclinic Bravais lattice mP.

Table 9.1.9.1| top | pdf |
Example

TransformationScalar products
121314232434
[{\bf b}_1][{\bf b}_2][{\bf b}_3][{\bf b}_4][-0.271][0.265][-22.02][-24.37][0.272][-32.51]
[-{\bf b}_1][{\bf b}_1 + {\bf b}_2][{\bf b}_3][{\bf b}_1 + {\bf b}_4][-21.75][-0.265][0][-24.10][\sim 0][-32.24]
[{\bf b}_1'][{\bf b}_3'][{\bf b}_4'][{\bf b}_2'][13][14][12][34][23][24]

The transformation related to this case leads to a monoclinic conventional cell but does not consider the possibility of shorter basis vectors. For this reason, it is necessary here to look at the other vectors of the set V in the ([{\bf b}_{1}'{\bf b}_{3}']) plane, the only one of interest is [{\bf b}_{1}' + {\bf b}_{3}']. The length of this vector is [4.936\ \hbox{\AA}], which is shorter than [{\bf b}_{3}'] ([|{\bf b}_{3}'| = 6.771\ \hbox{\AA}]) and leads to the cell parameters [a = 4.693, b = 5.678, c = 4.936\ \hbox{\AA}, \alpha = 90, \beta = 90.67, \gamma = 90^{\circ}].

References

First citation Azároff, L. V. & Buerger M. J. (1958). The powder method in X-ray crystallography. New York: McGraw-Hill.Google Scholar








































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